You trigger after you take damage but before you fall unconscious, like numerous other triggers in the game.

I fail to see how that's supported by RAW. If damage reduces you to 0 hit points and fails to kill you, you fall unconscious.

The fact that a Fighter subclass gives you an ability to act otherwise doesn't affect this, any more than you could say you stay conscious at 0 because Zealot Barbarians can do it. You don't have that ability. You're unconscious or dead when Vampiric Touch triggers. Vampiric Touch fails.